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Mukhanov-Sasaki equation in manifestly gauge-invariant linearized cosmological perturbation theory with dust reference fields

The aim of this article is to understand the role of dust reference fields, often also called clocks, on cosmological perturbations around a flat FLRW universe. We derive the Mukhanov-Sasaki (MS) equation for the Brown-Kuchar (BK) and Gaussian dust models, which both consider four dust fields as reference fields. The reduced phase space of Dirac observables, that is the gauge-invariant part of the theory, is constructed by means of an observable map applied to all elementary phase space variables of the coupled system, consisting of gravity, a massive scalar field and the dust degrees of freedom. The evolution of these observables is governed by a so called physical Hamiltonian which can be derived once the set of reference fields are chosen and differs for each model. First, the reduced phase space as well as the corresponding equations of motion are derived for full general relativity. Then from this, the gauge-invariant version of the equations of motion for the background are derived which contain a fingerprint of the dust reference fields. Afterwards we study linear cosmological perturbations around a FLRW metric using the scalar-vector-tensor decomposition and derive the equation of motion for the MS variable in this formalism for a chosen set of variables on the reduced phase space and expressed in terms of Dirac observables. The MS equation involves additional contributions that can be understood as back reactions from the dust reference fields. These additional dust contributions to the MS equation were absent if the dust energy and momentum density as well as their perturbations are vanishing. The nature of the correction terms suggest that BK and Gaussian dust reference fields contribute differently. Using numerical simulations we study the behavior of the dust contributions to the MS equation during inflation.